V. Ziegler

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math.NT2026

Product of powers of distinct primes as sums of Fibonacci numbers

Herbert Batte, Florian Luca, Volker Ziegler

Let $F_{n}$ be the $n$ -th Fibonacci number. In this paper, we study the Diophantine equation $F_{n} + F_{m} = p^{x} q^{y}$ in nonnegative integers $n \geq m$ , $x$ and $y$ , where $p$ and $q$ are fix…

math.NT2026

Rational integers as sums of units -- the quadratic case

Christopher Frei, Martin Widmer, Volker Ziegler

How many natural numbers below $X$ can be written as a sum of $k$ units of the ring of integers of a given number field? We give the asymptotics as $X$ gets large for quadratic num…

math.NT2023

On Pillai's Problem involving Lucas sequences of the second kind

Sebastian Heintze, Volker Ziegler

In this paper we consider the Diophantine equation $V_{n} - b^{m} = c$ for given integers $b, c$ with $b \geq 2$ , whereas $V_{n}$ varies among Lucas-Lehmer sequences of the second…

math.NT2023

On a conjecture of Levesque and Waldschmidt

Tobias Hilgart, Volker Ziegler

One of the first parametrised Thue equations, $X^{3} - (n - 1) X^{2} Y - (n + 2) X Y^{2} - Y^{3} = 1,$ over the integers was solved by E. Thomas in 1990. If we interpret this as…